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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 8. Global Analytic Functions }
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}
\item  Analytic Continuation
\begin{enumerate}
\item[1.1.] The Weierstrass Theory
\item[1.2.] Germs and Sheaves
\item[1.3.] Sections and Riemann Surfaces
\item[1.4.] Analytic Continuations along Arcs
\item[1.5.] Homotopic Curves
\item[1.6.] The Monodromy Theorem
\item[1.7.] Branch Points
\end{enumerate}

\item  Algebraic Functions
\begin{enumerate}
\item[2.1.] The Resultant of Two Polynomials
\item[2.2.] Definition and Properties of Algebraic Functions
\item[2.3.] Behavior at the Critical Points
\end{enumerate}
 

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3-4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item  Picard's Theorem
\begin{enumerate}
\item[3.1.] Lacunary Values
\end{enumerate}

\item  Linear Differential Equations
\begin{enumerate}
\item[4.1.] Ordinary Points
\item[4.2.] Regular Singular Points
\item[4.3.] Solutions at Infinity
\item[4.4.] The Hypergeometric Differential Equation
\item[4.5.] Riemann's Point of View
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{1.1. The Weierstrass Theory. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Weierstrass, in contrast to Riemann, who favored a more geometric outlook, wanted to build the whole theory of analytic functions from the concept of power series. 
%%
For Weierstrass the basic building block was a power series
$$
P(z-\zeta) = a_0 + a_1(z-\zeta) + \cdots + a_n(z-\zeta)^n + \cdots 
$$
with a positive radius of convergence $r(P)$. 
%%
The radius of convergence is given by Hadamard's formula 
$$
r(P)^{-1} = \varlimsup\limits_{n\to\infty} |a_n|^{1/n}. 
$$

}

\item  Answer. 
\begin{enumerate}
\item 
Such a series is determined by a complex number $\zeta$, the center of the power series, and a sequence $\{a_n\}_0^{\infty}$ of complex coefficients.


\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. Germs and Sheaves. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The Weierstrass theory has mostly historical interest, for the restriction to power series and their domains of convergence is more of a hindrance than a help. 
It should, nevertheless, be recognized that the idea of Weierstrass is still the basis for our understanding of multiple-valuedness in the theory of complex analytic functions.

}

\item  Answer. 
\begin{enumerate}
\item 
We shall outline a more direct approach which is more in line with the somewhat sophisticated ideas that dominate the recent theory of analytic functions of several complex variables.

\item 
An analytic function $f$ defined in a region $\Omega$ will constitute a function element, denoted by $(f,\Omega)$, and a global analytic function will appear as a collection of function elements which are related to each other in a prescribed manner.

%\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. Germs and Sheaves. Definition 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What is a sheaf? 
}

\item  Answer. 
A sheaf over $D$ is a topological space $\mathcal{S}$ and a mapping $\pi:\mathcal{S}\to D$ with the following properties:

\begin{enumerate}
\item  The mapping $\pi$ is a local homeomorphism; this shall mean that each $s\in\mathcal{S}$ has an open neighborhood $\Delta$ such that $\pi(\Delta)$ is open and the restriction of $\pi$ to $\Delta$ is a homeomorphism.

\item  For each $\zeta \in D$ the stalk $\pi^{-1}(\zeta) = \mathcal{S}_\zeta$ has the structure of an abelian group.

\item  The group operations are continuous in the topology of $\mathcal{S}$.

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Sections and Riemann Surfaces. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Let $\mathcal{S}$ be a sheaf over $D$ and consider an open set $U\subset D$. 
A continuous mapping $\varphi: U\to\mathcal{S}$ is called a section over $U$ if the composed mapping $\pi\circ\varphi$ is the identity mapping of $U$ on itself. 
%%
It follows from this condition that $\varphi(\zeta_1) = \varphi(\zeta_2)$ implies $\zeta_1 =\zeta_2$; hence $\varphi$ is one to one, and its inverse is $\pi$ restricted to $\varphi(U)$. Thus every section is a homeomorphism.

}

\item  Answer. 
\begin{enumerate}
\item 
In what follows $\mathcal{S}$ will always be the sheaf of germs of analytic functions over the whole complex plane. 

The components of $\mathcal{S}$, regarded as a topological space, can be identified with the global analytic functions.

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Analytic Continuation along Arcs. Theorem 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Two analytic continuations $\bar{\gamma}_1$ and $\bar{\gamma}_2$ of a global analytic function $f$ along the same arc $\gamma$ are either identical, or 
$\bar{\gamma}_1(t)\neq \bar{\gamma}_1(t)$ for all $t$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Analytic Continuation along Arcs. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If a function element is defined by a power series inside its circle of convergence, supposed to be of finite radius, prove that at least one radius is a singular path for the global analytic function which it determines. ("A power series has at least one singular point on its circle of convergence.")
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Analytic Continuation along Arcs. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If a function element $(f,\Omega)$ has no direct analytic continuations other than the ones obtained by restricting $f$ to a smaller region, then the boundary of $\Omega$ is called a natural boundary for $f$. Prove that the series $\sum\limits_{n=0}^{\infty} z^{n!}$ has the unit circle as a natural boundary. 
Hint: Show that the function tends to infinity on every radius whose argument is a rational multiple of $\pi$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Analytic Continuation along Arcs. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the function $\lambda(\tau)$ introduced in Chap. 7, Sec. 3.4, has the
real axis as a natural boundary.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.5. Homotopic Curves. Definition 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What is the definition of homotopic curves?
}

\item  Answer. 
Two arcs $\gamma_1$ and $\gamma_2$ over the same parameter interval $[a,b]$ are said to be homotopic in $\Omega$ if there exists a continuous function $\gamma(t,u)$, defined on a rectangle $[a,b] \times [0,1]$, with the following properties: 
\begin{enumerate}
\item $\gamma(t,u)\in\Omega$ for all $(t,u)$.
\item $\gamma(t,0) = \gamma_1(t)$, $\gamma(t,l) = \gamma_2(t)$ for all $t$.
\item $\gamma(a,u) = \gamma_1(a) = \gamma_2(a)$, $\gamma(b,u) = \gamma_1(b) = \gamma_2(b)$ for all $u$.

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.6. The Monodromy Theorem. Theorem 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If the arcs $\gamma_1$ and $\gamma_2$ are homotopic in $\Omega$, and if a given germ
of $f$ at the initial point can be continued along all arcs contained in $\Omega$, then the continuations of this germ along $\gamma_1$ and $\gamma_2$ lead to the same germ at the terminal point.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.7. Branch Points. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
For a closer study of the singularities of multiple-valued functions it is necessary to determine, explicitly, the fundamental group of a punctured disk.

Let the punctured disk be represented by $0 < |z| < \rho$, and consider a fixed point, for instance the point $z_0 = r < \rho$ on the positive radius. 

By means of a central projection, given by 
$$
\gamma(t,u) = (1-t)\gamma(t) + ur\frac{\gamma(t)}{|\gamma(t)|},
$$
any closed curve $\gamma$ from $z_0$ can be deformed into a curve which lies on the circle $|z| = r$. It is thus sufficient to consider curves on that circle. 
We continue to use the notation $\gamma(t)$.

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Resultant of Two Polynomials. Theorem 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $P(w,z)$ and $Q(w,z)$ are relatively prime polynomials, there are only a finite number of values $z_0$ for which the equations $P(w,z_0)=0$ and $Q(w,z_0)=0$ have a common root. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Definition and Properties of Algebraic Functions. Definition 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
When is a global analytic function said to be an algebraic function? 
}

\item  Answer. 
\begin{enumerate}
\item 
A global analytic function $f$ is called an algebraic function if all its function elements $(f,\Omega)$ satisfy a relation $P(f(z),z) = 0$ in $\Omega$, where $P(w,z)$ is a polynomial which does not vanish identically. 

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Behavior at the Critical Points. Theorem 4.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
An analytic function is an algebraic function if it has a finite number of branches and at most algebraic singularities. Every algebraic function $w = f(z)$ satisfies an irreducible equation $P(w,z) = 0$, unique up to a constant factor, and every such equation determines a corresponding algebraic function uniquely.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Behavior at the Critical Points. Exercise 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Determine the position and nature of the singularities of the algebraic function defined by $w^3 -3wz + 2z^3 = 0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.1. Lacunary Values. Theorem 5. Picard.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
An entire function with more than one finite lacunary value reduces to a constant.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.1. Ordinary Points. Theorem 6. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $z_0$ is an ordinary point for the equation
$$
a_0(z)w'' + a_1(z)w' + a_2(z)w = 0, 
$$
there exists a local solution $(f,\Omega), z_0 \in \Omega$, with arbitrarily described values $f(z_0)=b_0$ and $f'(z_0)=b_1$. The germ $(f,z_0)$ is uniquely determined.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.1. Ordinary Points. Exercise 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Find the power-series developments about the origin of two linearly independent solutions of $w'' = zw$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.1. Ordinary Points. Exercise 2.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The Hermite polynomials are defined by 
$$H_n(z) = (-1)^ne^{z^2}\frac{d^n}{dz^n}(e^{-z^2}). $$
Prove that $H_n(z)$ is a solution of $w''-2zw' + 2nw = 0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.2. Regular Singular Points. Theorem 7. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $z_0$ is a regular singular point for the equation (10), there exist linearly independent solutions of the form $(z-z_0)^{\alpha_1}g_1(z)$ and $(z-z_0)^{\alpha_2}g_2(z)$ with $g_1(0),g_2(0)\neq 0$ corresponding to the roots of the indicial equation, provided that $\alpha_2-\alpha_1$ is not an integer. 
In the case of an integral difference $\alpha_2-\alpha_1\ge 0$ the existence of a solution corresponding to $\alpha_2$ can still be asserted. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.2. Regular Singular Points. Exercise 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the equation 
$$(1-z^2)w''-2zw'+n(n+1)w = 0,$$
where $n$ is a nonnegative integer, has the Legendre polynomials
$$ P_n(z) = \frac{1}{2^nn!}\cdot \frac{d^n}{dz^n}(z^2-1)^n $$
as solutions.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.2. Regular Singular Points. Exercise 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Determine two linearly independent solutions of the equation
$$z^2(z+1)w'' - z^2w' + w = 0$$
near $0$ and one near $-1$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.2. Regular Singular Points. Exercise 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that Bessel's equation $zw'' + w' + zw = 0$ has a solution which is an integral function. Determine its power-series development.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.3. Solutions at Infinity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $a_0(z)$, $a_1(z)$, $a_2(z)$ are polynomials, it is natural to ask how the solutions behave in the neighborhood of $\infty$. 
}

\item  Answer. 
\begin{enumerate}
\item 
The most convenient way to treat this question is to make the variable transformation $z = 1/Z$. 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Hypergeometric Differential Equation. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
We have just seen that differential equations with one or two regular singularities have trivial solutions. It is only with the introduction of a third singularity
that we obtain a new and interesting class of analytic functions.
}

\item  Answer. 
\begin{enumerate}
\item 
It is quite clear that a linear transformation of the variable transforms a second-order linear differential equation into one of the same type and that the character of the singularities remains the same.

\item 
We can therefore elect to place the three singularities at prescribed points, and it is simplest to choose them at $0$, $1$, and $\infty$. 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Hypergeometric Differential Equation. Exercise 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that $(1-z)^{-\alpha} = F(\alpha,\beta,\beta,z)$ and 
$\log 1/(1- z) = zF(1,1,2,z)$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Hypergeometric Differential Equation. Exercise 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Express the derivative of $F(a,b,c,z)$ as a hypergeometric function. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Hypergeometric Differential Equation. Exercise 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Derive the integral representation
$$
F(a,b,c,z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} 
\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Hypergeometric Differential Equation. Exercise 4. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $w_1$ and $w_2$ are linearly independent solutions of the differential equation $w'' = pw' + qw$, prove that the quotient $\eta = w_2/w_1$ satisfies
$$
\frac{d}{dz}\left( \frac{\eta''}{\eta'} \right) 
- \frac{1}{2}\left( \frac{\eta''}{\eta'} \right)^2 
= -2q - \frac{1}{2}p^2 +p'. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.5. Riemann's Point of View. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Riemann was a strong proponent of the idea that an analytic function can be defined by its singularities and general properties just as well as or perhaps better than through an explicit expression.
}

\item  Answer. 
\begin{enumerate}
\item 
A trivial example is the determination of a rational function by the singular parts connected with its poles.
 
\item 
We will show, with Riemann, that the solutions of a hypergeometric differential equation can be characterized by properties of this nature.

\item 
We consider in the following a collection $\mathcal{F}$ of function elements $(f,\Omega)$ with certain characteristic features which we proceed to enumerate. 
 
\end{enumerate}

\end{itemize}

\end{frame}

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\end{document}

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